A 330 ml beverage can
Cylinder Volume and Material Cost
The 330 ml on the can is not advertising, it is a legally binding promise: 330 ml has to come out of it. The maker cannot touch that number. The only thing left to change is the shape — and that isn't free either, because metal is bought by the square centimetre. Infinitely many cylinders hold the same 330 ml. They do not all cost the same.
Three ways to see it
The volume inside is fixed: 330 ml, which is 330 cm³. The metal outside is a film — the wall about 0.1 mm, the lid roughly three times that, because there is pressurised gas inside. Aluminium's density is 2.7 g/cm³, so one extra cm² of surface is only 0.027 g per can — and 27 tonnes across a billion cans. That is why the factory watches the surface this closely. Look at the fill line too: the machine cannot hit 330 exactly, so it always aims a little high. Coming up short is illegal; going over is merely expensive.
Switch layers — the scene stays put
What's really going on
A fixed volume does not fix the shape, but it takes one freedom away: the instant you choose r, h is no longer yours. One question is left — which shape wraps the same contents in the least skin? The scaling law answers it: enlarge an object and volume grows with the cube of size while surface grows with the square, so A/V falls. That is why a 1 litre can spends about a third less metal per millilitre, and why nobody sells a 10 ml can. The same law also picks the shape: a sphere wraps a given volume in the smallest surface, and among cylinders the one nearest a sphere is h = 2r. The real can's departure from that is not sloppiness — it is a new piece of information entering the arithmetic: the lid is thicker, so area at the ends is dearer, and the cheapest can grows narrower and taller. Geometry does not decide alone; it decides once you tell it what each square centimetre costs.
The equation
V = π r² h · A = 2πr² + 2πrh
Volume: the base circle has area πr², and stacking it h high gives πr²h. The surface arrives in two parts — the disc at the bottom and the one on top (2 × πr²), and the side. To see the side, cut the can along a vertical line and unroll it: it becomes a plain rectangle of height h and width 2πr, the circle's circumference, so its area is 2πrh. Once V is fixed, h = V/(πr²), and A depends on r alone.
Same concept, elsewhere
Memorising one example gets you nowhere. You need to spot the concept wherever it turns up.
A food tin — same fixed volume, same surface arithmetic; its height and its diameter are both set by material cost
A farm water tank — 10 m³ = 10,000 litres; volume grows with the cube of size while sheet area grows with the square, so a big tank needs less metal per litre
A finned radiator — here the goal is inverted: the most surface possible from the same volume, because heat leaves through the surface
A soap bubble — a sphere wraps a given volume in the least surface; that is the limit a can cannot reach, because a can has to stand upright and stack
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